Abstract: We have calculated the next-to-leading order cross sections for the inclusiveproduction of D*-mesons in ep collisions at HERA for finite, although verysmall Q2. In this Q2-range, the same approximations as for photoproduction canbe used. Our calculation is performed in the general-massvariable-flavour-number scheme. In this approach, large logarithms of the charmtransverse momentum are resummed and finite terms depending on m^2-p T^2 arekept in the hard scattering cross sections. The theoretical results arecompared with recent data from the ZEUS collaboration at HERA. On average, wefind good agreement.